1) Start building it ahead of time. If you get invited to a party on Monday, and it takes place on Saturday, you're going to want to tell people you're aunt died on Wednesday. If she dies on Saturday morning, they're not going to have the funeral that night.
2) Choose your sickness wisely. Instead of saying you have a headache, say you have "migraines." It's not a sore throat, it's "strep." Food poisoning is a good sickness, but be sure to say the specific food that made you sick and add "I knew it would happen, too. I don't know why I keep eating there."
3) Try to blame the excuse on someone else. "I'd love to go, but that asshole boss of mine is making me work tonight." It sounds a lot better than "I'd love to go, but I have to work." That sounds like you made it up.
4) The more elaborate, the better. At least, to a degree. "I couldn't make it on time because I had a flat tire, and I called AAA, but they took forever to get here because there was an accident" sounds more believable than "my car broke down and I tried to walk here in time, but I was mugged."
5) Use what people know about you as part of your story. If you're a big sports fan, you can say you can't go to a wedding because you bought tickets to a big game in another city, and you already paid for the hotel and everything. Oddly enough, the more people know about you, the easier it is to lie to them.
6) If for some reason you have procrastinated on your excuse making, and need to lie at the last possible second, say you're on you're way there, but you got lost. Then, call every thirty minutes saying you're still lost, but you think you've found your way back onto the highway. Yes, you will look like an idiot, but you also won't have to watch your friend's stupid kid play the piano, or whatever the hell they invited you to go do.